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A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

%I A053156 #34 Jan 29 2023 19:36:50
%S A053156 1,3,10,33,106,333,1030,3153,9586,29013,87550,263673,793066,2383293,
%T A053156 7158070,21490593,64504546,193579173,580868590,1742867913,5229128026,
%U A053156 15688432653,47067395110,141206379633,423627527506,1270899359733
%N A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.
%C A053156 Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - _Ross La Haye_, Jan 12 2008
%C A053156 From _Paul Barry_, Apr 27 2003: (Start)
%C A053156 With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
%C A053156 G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
%C A053156 E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
%C A053156 Binomial transform of A083329.
%C A053156 Second binomial transform of A040001. (End)
%H A053156 G. C. Greubel, <a href="/A053156/b053156.txt">Table of n, a(n) for n = 1..1000</a>
%H A053156 V. Jovovic, G. Kilibarda, <a href="http://dx.doi.org/10.4213/dm398">On the number of Boolean functions in the Post classes F^{mu}_8</a>, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
%H A053156 V. Jovovic, G. Kilibarda, <a href="http://dx.doi.org/10.1515/dma.1999.9.6.593">On the number of Boolean functions in the Post classes F^{mu}_8</a>, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
%H A053156 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A053156 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A053156 a(n) = (3^n - 2^n + 1)/2.
%F A053156 a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - _Ross La Haye_, Jan 12 2008
%F A053156 From _Colin Barker_, Jul 29 2012: (Start)
%F A053156 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
%F A053156 G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)
%p A053156 A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # _Wesley Ivan Hurt_, Oct 06 2017
%t A053156 LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* _G. C. Greubel_, Oct 06 2017 *)
%o A053156 (PARI) a(n) = (3^n-2^n+1)/2; \\ _Michel Marcus_, Nov 30 2015
%o A053156 (Magma) [(3^n-2^n+1)/2: n in [1..30]]; // _G. C. Greubel_, Oct 06 2017
%Y A053156 Cf. A000225, A000392, A028243, A000079.
%Y A053156 Cf. A036239.
%Y A053156 Column k=2 of A288638.
%Y A053156 Third column of A294201.
%K A053156 easy,nonn
%O A053156 1,2
%A A053156 _Vladeta Jovovic_ and Goran Kilibarda, Feb 28 2000